Extension of Delaunay normalisation for arbitrary powers of the radial distance

Abstract

In the framework of perturbed Keplerian systems we deal with the Delaunay normalisation of a wide class of perturbations such that the radial distance is raised to an arbitrary real number $\gamma$. The averaged function is expressed in terms of the Gauss hypergeometric function ${}_2{F}_1$ whereas the associated generating function is the so called Appell hypergeometric function $F_1$. The Gauss hypergeometric function related to the average depends on the eccentricity, $e$, whereas the Appell function depends additionally on the eccentric anomaly, $E$, and both special functions are properly defined and evaluated for all $e\in [0,1)$ and $E\in [-\pi ,\pi ]$. We analyse when the functions we determine can be extended to $e=1$. When the exponent of the radial distance is an integer, the usual values of the averaged and generating functions are recovered.